A directed graph (digraph) is formed by vertices and arcs (directed edges) from one vertex to another. A feedback vertex set (FVS) is a set of vertices that contains at least one vertex of every directed cycle in this digraph. The directed feedback vertex set problem aims at constructing a FVS of minimum cardinality. This is a fundamental cycle-constrained hard combinatorial optimization problem with wide practical applications. In this paper we construct a spin glass model for the directed FVS problem by converting the global cycle constraints into local arc constraints, and study this model through the replica-symmetric (RS) mean field theory of statistical physics. We then implement a belief propagation-guided decimation (BPD) algorithm for single digraph instances. The BPD algorithm slightly outperforms the simulated annealing algorithm on large random graph instances. The RS mean field results and algorithmic results can be further improved by working on a more restrictive (and more difficult) spin glass model.

National Basic Research Program of China [2013CB932804]
; National Basic Research Program of China [2013CB932804]
; National Basic Research Program of China [2013CB932804]
; National Basic Research Program of China [2013CB932804]
; National Natural Science Foundation of China [11121403, 11225526]
; National Natural Science Foundation of China [11121403, 11225526]
; National Natural Science Foundation of China [11121403, 11225526]
; National Natural Science Foundation of China [11121403, 11225526]
; Knowledge Innovation Program of Chinese Academy of Sciences [KJCX2-EW-J02]
; Knowledge Innovation Program of Chinese Academy of Sciences [KJCX2-EW-J02]
; Knowledge Innovation Program of Chinese Academy of Sciences [KJCX2-EW-J02]
; Knowledge Innovation Program of Chinese Academy of Sciences [KJCX2-EW-J02]
; National Basic Research Program of China [2013CB932804]
; National Basic Research Program of China [2013CB932804]
; National Basic Research Program of China [2013CB932804]
; National Basic Research Program of China [2013CB932804]
; National Natural Science Foundation of China [11121403, 11225526]
; National Natural Science Foundation of China [11121403, 11225526]
; National Natural Science Foundation of China [11121403, 11225526]
; National Natural Science Foundation of China [11121403, 11225526]
; Knowledge Innovation Program of Chinese Academy of Sciences [KJCX2-EW-J02]
; Knowledge Innovation Program of Chinese Academy of Sciences [KJCX2-EW-J02]
; Knowledge Innovation Program of Chinese Academy of Sciences [KJCX2-EW-J02]
; Knowledge Innovation Program of Chinese Academy of Sciences [KJCX2-EW-J02]

National Basic Research Program of China [2013CB932804]
; National Basic Research Program of China [2013CB932804]
; National Basic Research Program of China [2013CB932804]
; National Basic Research Program of China [2013CB932804]
; National Natural Science Foundation of China [11121403, 11225526]
; National Natural Science Foundation of China [11121403, 11225526]
; National Natural Science Foundation of China [11121403, 11225526]
; National Natural Science Foundation of China [11121403, 11225526]
; Knowledge Innovation Program of Chinese Academy of Sciences [KJCX2-EW-J02]
; Knowledge Innovation Program of Chinese Academy of Sciences [KJCX2-EW-J02]
; Knowledge Innovation Program of Chinese Academy of Sciences [KJCX2-EW-J02]
; Knowledge Innovation Program of Chinese Academy of Sciences [KJCX2-EW-J02]
; National Basic Research Program of China [2013CB932804]
; National Basic Research Program of China [2013CB932804]
; National Basic Research Program of China [2013CB932804]
; National Basic Research Program of China [2013CB932804]
; National Natural Science Foundation of China [11121403, 11225526]
; National Natural Science Foundation of China [11121403, 11225526]
; National Natural Science Foundation of China [11121403, 11225526]
; National Natural Science Foundation of China [11121403, 11225526]
; Knowledge Innovation Program of Chinese Academy of Sciences [KJCX2-EW-J02]
; Knowledge Innovation Program of Chinese Academy of Sciences [KJCX2-EW-J02]
; Knowledge Innovation Program of Chinese Academy of Sciences [KJCX2-EW-J02]
; Knowledge Innovation Program of Chinese Academy of Sciences [KJCX2-EW-J02]